Is the Grothendieck group finite degree generated?

Question

For a scheme $X$ with structure sheaf $\mathcal{O}_X$, is the Grothendieck ring of locally free $\mathcal{O}_X$-modules generated by the equivalence classes of rank $≤N$ modules, where $N$ is a sufficiently large integer? Is it true when $X$ is projective or furthermore $X$ is a projective variety?

Answer 1

If $X$ is quasiprojective, the ring $K_0(X)$ is generated by vector bundles of rank at most $\dim(X)$.

Indeed, if $E$ is a vector bundle of rank $r > \dim(X)$ and $L$ is an ample line bundle, tensor product $E \otimes L^n$ is globally generated for $n \gg 0$, hence a general global section of $E$ has no zeroes, and hence there is an exact sequence $$ 0 \to L^{-n} \to E \to E' \to 0, $$ where $E'$ is locally free of rank $r - 1$, and hence $$ [E] = [L^{-n}] + [E']. $$